A separative power of a separating element, whose heads and tails separation factors are a and g, is expressed by ψ
b (a, β)=[α(β-1)In α-(α-1)In β]/(αβ-1) for the unit flow of the desired material and ψ
a(α, β)(≡ψ
b(β, α)) for that of undesired material. The additive properties of the functions ψ
b and ψ
a, were demonstrated by calculations of various types of ideal cascades, but the origin of the property is not obvious. The present study has furnished the mathematical basis of the additivity based on the special functional equa-tion. First, for symmetric processes (α=β), the functional equation which describes the function representing the quality of separation
f(α, α) concerning the desired material was obtained and solved to give the functional form of
f(α, α). The result was extented to the function
f(α, β) representing the quality of asymmetric separation (α≠β). The derived function
f(α, β) was demonstrated to be equal to ψ
b(α, β), and it was verified that functions ψ
b(α, β) and ψ
a(α, β) have the additive property in themselves.
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